PHYSICAL REVIEW E 70, 061804 (2004)
Polyelectrolyte adsorption: Chemical and electrostatic interactions
Adi Shafir* and David Andelman†
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,
Ramat Aviv, Tel Aviv 69978, Israel
(Received 3 December 2003; revised manuscript received 24 August 2004; published 28 December 2004)
Mean-field theory is used to model polyelectrolyte adsorption and the possibility of overcompensation of
charged surfaces. For charged surfaces that are also chemically attractive, the overcharging is large in high salt
conditions, amounting to 20–40 % of the bare surface charge. However, full charge inversion is not obtained in
thermodynamical equilibrium for physical values of the parameters. The overcharging increases with addition
of salt, but does not have a simple scaling form with the bare surface charge. Our results indicate that a more
evolved explanation is needed in order to understand polyelectrolyte multilayer buildup. For strong polymerrepulsive surfaces, we derive simple scaling laws for the polyelectrolyte adsorption and overcharging. We show
that the overcharging scales linearly with the bare surface charge, but its magnitude is very small in comparison
to the surface charge. In contrast with the attractive surface, here the overcharging is found to decrease
substantially with addition of salt. In the intermediate range of weak repulsive surfaces, the behavior with
addition of salt crosses over from increasing overcharging (at low ionic strength) to a decreasing one (at high
ionic strength). Our results for all types of surfaces are supported by full numerical solutions of the mean-field
equations.
DOI: 10.1103/PhysRevE.70.061804
PACS number(s): 82.35.Gh, 82.35.Rs, 61.41.⫹e
I. INTRODUCTION
Aqueous solutions containing polyelectrolytes and small
ions are abundant in biological systems, and have been the
subject of extensive research in recent years. When such a
solution is in contact with an oppositely charged surface,
adsorption of the polyelectrolyte chains can occur. Theoretical descriptions of polyelectrolyte adsorption take into account the multitude of different interactions and length
scales. Among others they include electrostatic interactions
between the surface, monomers, and salt ions, excludedvolume interactions between monomers and entropy considerations. Although a full description of polyelectrolytes is
still lacking at present, several approaches exist and use different types of approximations [1–23]. These include linearized mean-field equations [1–6], numerical solutions of nonlinear mean-field equations [7–10], scaling considerations
[7–16], multi-Stern layers of discrete lattice models [17–20],
and computer simulations [21–23].
Experimental studies [24–26] have shown that adsorbing
polyelectrolytes (PE’s) may carry a charge greater than that
of the bare surface, so that the overall surface-polyelectrolyte
complex has a charge opposite to that of the bare charged
surface. This phenomenon is known as overcharging (or surface charge overcompensation) by the PE chains. When the
overcharging is large enough to completely reverse the bare
surface charge, the resulting charge surplus of the complex
can be used to attract a second type of polyelectrolyte having
an opposite charge to that of the first polyelectrolyte layer.
Eventually, by repeating this process, a complex structure of
alternating layers of positively and negatively charged poly-
*Electronic address: [email protected]
†
Electronic address: [email protected]
1539-3755/2004/70(6)/061804(12)/$22.50
electrolytes can be formed. Experimentally, multilayers consisting of hundreds of such layers can be created [24,25],
leading the way to several interesting applications.
A theoretical description of the PE overcharging was proposed in Refs. [6,27,28] based on a mean-field formalism.
The model relies on several approximations for a very dilute
PE solution in contact with a charged surface and in theta
solvent condition. In the high salt limit, the model predicts
an exact charge inversion for indifferent (i.e., noninteracting)
surfaces. In another work [7], the scaling of the adsorption
parameters was derived using a Flory-like free energy. This
work used mean-field theory, but with a different type of
boundary conditions. Using a strongly (nonelectrostatic) repulsive surface for the PE, profiles of monomer concentration and electrostatic potential have been calculated from numerical solution of the mean-field equations. Several scaling
laws have been proposed and the possibility of a weak overcharging in low salt condition has been demonstrated. In a
related work [10] we presented simple scaling laws resulting
from the same mean-field equations and for the same chemically repulsive surfaces. In particular, we addressed the
adsorption-depletion crossover as function of added salt. We
showed that addition of salt eventually causes the polyelectrolyte to deplete from the charged surface, and pre-empts
the high-salt adsorption regime described in Ref. [7].
The present paper can be regarded as a sequel to Ref.
[10], offering a more complete treatment of the adsorption
problem for several types of charged surfaces. In particular,
it includes the effect of surface-PE chemical interactions, and
the scaling of overcharging and adsorption in the presence of
added salt. These chemical interactions are found to play a
crucial role in the overcharging, showing that a necessary
condition for the formation of multilayers is a chemical attraction of nonelectrostatic origin (complexation) between
the two types of PE chains as well as between the PE and the
charged surface. For chemically attractive surfaces our re-
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©2004 The American Physical Society
PHYSICAL REVIEW E 70, 061804 (2004)
A. SHAFIR AND D. ANDELMAN
sults deviate from those of Refs. [6,27,28]. We find in the
same solvent and surface conditions that the overcharging
does not reach a full 100% charge inversion of the bare surface charge. It rather depends on system parameters and
never exceeds 30–40 % for the physical range of parameters.
Like in Ref. [6], we find an increase of the overcharging with
salt but our numerical results do not agree with the previous
prediction. For repulsive surfaces, several scaling laws are
obtained and agree well with the numerically obtained profiles. However, due to the competition between the electrostatic and chemical surface interactions, the overcharging
here is usually quite small, of the order of ⬃1% only.
The paper is organized as follows: the mean-field equations, used in the past in several other works for PE adsorption, are reviewed in Sec. II. The following two sections treat
two different types of surfaces. In Sec. III, results for PE
adsorption and overcharging for attractive surfaces are presented and compared with previous models. In Sec. IV we
derive the scaling of adsorbed PE layers in the case of a
chemically repulsive surface. In Sec. V we present the adsorption and overcharging for the intermediate case of a
weakly repulsive surface. In particular, we show the dependence of PE adsorption on the solution ionic strength. A
summary of the main results and future prospects are presented in Sec. VI.
II. MEAN-FIELD EQUATIONS AND SURFACE
CHARACTERISTICS
Consider an aqueous solution, containing a bulk concentration of infinitely long polyelectrolyte (PE) chains, together
with their counterions and a bulk concentration of salt ions.
Throughout this paper we assume for simplicity that the PE’s
are positively charged and that the counterions and added
salt are all monovalent. The mean-field equations describing
such an ionic solution have been derived elsewhere [7,10]
and are briefly reviewed here:
ⵜ 2␺ =
8␲ecsalt
4 ␲ e 2 ␤e␺
共␾b fe − f ␾2兲,
sinh共␤e␺兲 +
⑀
␧
a2 2
ⵜ ␾ = v共␾3 − ␾2b␾兲 + ␤ fe␺␾ ,
6
共1兲
共2兲
where the polymer order parameter ␾共r兲 is the square root of
c共r兲, the local monomer concentration, ␾2b the bulk monomer
concentration, ␺ the local electrostatic potential, csalt the bulk
salt concentration, ␧ the dielectric constant of the aqueous
solution, f the monomer charged fraction, e the electron
charge, v the second virial (excluded volume) coefficient of
the monomers, a the monomer size, and ␤ = 1 / kBT the inverse of the thermal energy. Equation (1) is the PoissonBoltzmann equation where the salt ions, counterions, and
monomers are regarded as sources of the electrostatic potential. Equation (2) is the mean-field (Edwards) equation for
the polymer order parameter ␾共r兲, where the excludedvolume interaction between monomers and external electrostatic potential ␺共r兲 are taken into account.
For the case of an infinite planar wall at x = 0, Eqs. (1) and
(2) can be transformed into two coupled ordinary differential
equations, which depend only on the distance x from the
surface,
d 2␨
2 ␨
= ␬2 sinh ␨ + km
共e − ␩2兲,
dx2
共3兲
a 2 d 2␩
= v␾2b共␩3 − ␩兲 + f ␨␩ ,
6 dx2
共4兲
where ␨ ⬅ e␺ / kBT is the dimensionless (rescaled) electrostatic potential, ␩2 ⬅ ␾2 / ␾2b the dimensionless monomer concentration, ␬−1 = 共8␲lBcsalt兲−1/2 the Debye-Hückel screening
−1
= 共4␲lB␾2b f兲−1/2 a
length, due to added salt concentration, km
similar decay length due to counterions, and lB = e2 / ␧kBT the
Bjerrum length. For water with dielectric constant ␧ = 80 at
room temperature, lB is equal to about 7 Å. Note that the
actual decay of the electrostatic potential is determined by a
combination of salt, counterions, and polymer screening effects.
The solution of Eqs. (3) and (4) requires four boundary
conditions for the two profiles. The electrostatic potential
decays to zero at infinity, ␨共x → ⬁兲 = 0. At the surface, we
have chosen to work with a constant surface potential.
Namely, a conducting surface with a potential ␺ = ␺s, or in
rescaled variables, ␨共0兲 = −兩␨s兩. The results can be easily extended to the case of fixed surface charge density (which
amounts to fixing the derivative of the potential 兩d␨ / dx兩x=0
= −4␲lB␴).
For the PE profile we also have two boundary conditions.
At infinity, ␩共x → ⬁兲 = 1, because ␾共⬁兲 = ␾b has to match the
bulk value. The special case of a zero bulk monomer concentration can be treated by taking ␾b → 0 and working directly with the nonrescaled concentration ␾共x兲.
We model separately two types of surfaces. Although the
surface always attracts the oppositely charged PE chains, it
can be either chemically repulsive or attractive. In the case of
a chemical repulsion between the PE chains and the surface,
the amount of PE chains in direct contact with the surface is
set to a small value ␩共x = 0兲 = ␩s. In the limit of a strong
repulsive surface, ␩s tends to zero and the boundary condition is ␩共x = 0兲 = 0. Because of the ever-present longer-range
electrostatic attraction with the surface, PE chains accumulate in the surface vicinity, resulting in a positive slope
d␩ / dx 兩 x=0 ⬎ 0.
For chemical attractive surfaces we rely on a boundary
condition often used [6,29] for neutral polymer chains:
d␩共0兲 / dx + ␩共0兲 · d−1 = 0 where d has units of length and is
inversely proportional to the strength of nonelectrostatic interactions of the PE’s with the surface. The limit of d → ⬁ is
the indifferent (noninteracting) surface limit, while the previous limit of a strong repulsive surface is obtained by a
small and negative d.
In the following sections we present our results, first for
the chemically attractive surface and then for the chemically
repulsive one.
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POLYELECTROLYTE ADSORPTION: CHEMICAL AND …
PHYSICAL REVIEW E 70, 061804 (2004)
III. OVERCHARGING OF CHEMICALLY ATTRACTIVE
SURFACES
In this section we restrict the attention to surfaces that
attract the polymer in a nonelectrostatic (short range) fashion. For example, a possible realization can be a system containing PE chains with hydrophobic groups (like polystyrene
sulfonate) that are attracted to a hydrophobic surface, like the
water-air interface. The short range attraction is modeled by
a surface interaction in a similar way that is used extensively
for neutral polymers via a boundary condition on the polymer order parameter at the surface x = 0:
冏
␩共0兲 = − d ·
d␩
dx
冏
,
共5兲
x=0
where the length d was introduced in Sec. II and is inversely
proportional to the strength of the surface interaction.
Close to a surface the polymer has a concentration profile
given in terms of the distance from the surface x. Within
mean-field theory the adsorbed polymer amount is defined
with respect to the bulk concentration ␾2b to be
⌫⬅
冕
⬁
0
dx共␾2 − ␾2b兲 = ␾2b
冕
⬁
dx共␩2 − 1兲.
共6兲
0
We note that the definition of ⌫ manifests one of the deficiencies of mean-field theory where it is not possible to distinguish between chains absorbed on the surface and those
accumulated in the surface vicinity. The latter will be washed
away when the surface is placed in a clean aqueous solution
and does not participate in the effective PE surface buildup.
Another important quantity to be used throughout this paper is the overcharging parameter defined as the excess of PE
charge over that of the bare surface charge per unit area, ␴:
⌬␴ ⬅ − 兩␴兩 + f⌫,
共7兲
where ␴ ⬍ 0 is the induced surface charge density calculated
from the fixed surface potential 共␨⬘ = −4␲lB␴兲 in units of e,
the electron charge. Note that in terms of the relative overcharging parameter ⌬␴ / 兩␴兩, −1 艋 ⌬␴ / 兩␴兩 艋 0 corresponds to
undercharging, while a positive ⌬␴ / 兩␴兩 ⬎ 0 to overcharging.
The special value ⌬␴ / 兩␴兩 = 1 indicates a full charge inversion.
We solved numerically the mean-field coupled equations,
Eqs. (3) and (4), with the electrostatic boundary condition of
a fixed surface potential 兩␨s兩 (that has a one-to-one correspondence with an induced surface charge density ␴) and a nonelectrostatic boundary condition from Eq. (5). The relative
overcharging ⌬␴ / 兩␴兩 is plotted in Fig. 1 as function of the
amount of salt. The solid line represents our numerical results, while the dashed one corresponds to a previous prediction [6]. The same system parameters are used for both: the
limit of a dilute PE reservoir 共␾b = 0兲, theta solvent conditions 共v = 0兲, a strong ionic strength, and an indifferent surface taken in the limit of d → ⬁ (obtained already for d
艌 100 Å). Figure 1 shows an increasing dependence of the
overcharging parameter on csalt, but does not obey any
simple scaling law. It varies on quite a large range of values
from less than 10% (relative to 兩␴兩) for csalt ⯝ 0.1M to about
FIG. 1. The relative overcharging, ⌬␴ / 兩␴兩 = 共f⌫ − 兩␴兩兲 / 兩␴兩, is presented as a function of the amount of added salt csalt. The solid line
corresponds to the numerical results (Sec. III), while the dashed line
corresponds to the predictions from Eq. (9) in Ref. [6]. As the
surface potential 兩␨s兩 is fixed, ␴ is the numerically calculated induced surface charge. The numerical results show a much lower
overcharging than the predicted ones. Furthermore, the salt dependence of the overcharging is shown to be much stronger. The system parameters have been chosen to match those of Ref. [6]. A
dilute aqueous solution 共␾b = 0兲, in theta solvent conditions 共v = 0兲 is
used. Other parameters are ␧ = 80, T = 300 K, 兩␨s兩 = 1.0, a = 5 Å, f
= 1, d = 100 Å.
55% for csalt ⯝ 5.5M. We never observed for reasonable values of the parameters a full charge inversion of 100% or
more.
Our results are contrasted with the approximated prediction of Ref. [6]. In the high salt limit the prediction reads
2a2
⌬␴
=1+
csalt ,
兩␴兩
3df兩␴兩
共8兲
where we note that in Ref. [6] all lengths are rescaled with
a / 冑6. This prediction is shown by the dashed line in Fig. 1.
One can see that the linear dependence on salt concentration
is very weak and that for the entire range of salt the result is
dominated by the first and constant term in Eq. (8), giving an
overcharging of 100–110 %. Although the general trend of
an increase in ⌬␴ / 兩␴兩 appears in both results, there is neither
agreement in the values of ⌬␴ / 兩␴兩 nor in the quantitative
dependence on csalt for a wide range of csalt values.
The results of Fig. 1 have been done in the limit of an
indifferent surface modeled by d Ⰷ ␬−1. A closer examination
of the overcharging d dependence is presented in Fig. 2.
Both in part (a) for f = 0.2 and in part (b) for f = 1, our result
and the prediction of Eq. (8) show a similar descending trend
with d. Already for d as small as 100 Å the asymptotic results for the indifferent surface result is obtained. For smaller
d, the chemical attraction causes a bigger overcharging. The
main discrepancy between our exact solution of the meanfield equations and Eq. (8) is in the actual limiting value for
d → ⬁ and the lack of charge inversion from our results
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A. SHAFIR AND D. ANDELMAN
FIG. 2. The relative overcharging, ⌬␴ / 兩␴兩, is plotted against the chemical interaction parameter d, for (a) f = 0.2 and (b) f = 1. The solid
line corresponds to the numerically calculated relative overcharging, while the dashed line corresponds to the theoretical predictions taken
from Ref. [6]. Unlike the previous prediction, we do not observe a full charge inversion. The calculations are done in high salt conditions
csalt = 1M, corresponding to a Debye length of about 3 Å. Other parameters used are the same as in Fig. 1. Although the numerical and
predicted profiles show a similar qualitative dependence with d, they do not coincide, and converge to different values at d → ⬁. For practical
purposes, d = 100 Å is already a very good approximation for the indifferent d → ⬁ surface.
(solid line in Fig. 2). Note also that the difference between
the two results cannot be explained by a constant multiplicative factor, and that ⌬␴ / 兩␴兩 is smaller for f = 1 than for f
= 0.2 due to the larger electrostatic repulsion between
charged monomers.
To complete the presentation of the attractive surface we
show in Fig. 3 the dependence of the overcharging on csalt for
several different values of surface potential, ranging from
兩␨s兩 = 1 to 0.2. In part (a) the PE absorbed amount ⌫ (in units
of Å−2) is shown while in part (b) the dimensionless ⌬␴ / 兩␴兩
is shown. The general trend is an increase of both quantities
with csalt due to the nonelectrostatic attraction of the PE
chain to the surface on one hand and the screening of the
monomer-monomer repulsion on the other. The overcharging
FIG. 3. (a) The numerically calculated adsorbed amount ⌫ = 兰⬁0 dx共␾2 − ␾2b兲 is plotted as a function of the amount of added salt for three
values of the surface potential 兩␨s兩. The solid line corresponds to 兩␨s兩 = 1.0, the dashed-dotted line to 兩␨s兩 = 0.5, and the dashed line to 兩␨s兩
= 0.2. The adsorbed amount is shown to increase both with the amount of added salt and with 兩␨s兩. Other parameters used are as in Fig. 1.
In (b) the same results are plotted for ⌬␴ / 兩␴兩 as a function of the amount of added salt. The relative overcharging ⌬␴ / 兩␴兩 is seen to decrease
with 兩␨s兩, in contrast to ⌫ where it increases [see part (a)]. This implies that the surface charge ␴ has a stronger dependence on 兩␨s兩 than the
adsorbed amount ⌫. Therefore, the adsorbed amount no longer scales with ␴ for attractive surfaces, in contrast to repulsive surfaces,
Eq. (18).
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PHYSICAL REVIEW E 70, 061804 (2004)
is less than 100% unless the amount of salt is unrealistically
large. Another observation can be seen from the behavior of
⌬␴ / 兩␴兩. We clearly see that ⌬␴ does not depend linearly on
␴ since the three different 兩␨s兩 give three different curves (no
data collapse).
This very last result should be compared with the chemically repulsive surfaces which is discussed next and for
which we find that ⌬␴ ⬃ ␴.
IV. STRONG CHEMICALLY REPULSIVE SURFACES
The chemical repulsion between the PE chains and the
surface causes the amount of PE chains in direct contact with
the surface to diminish or even be zero for the strong repulsive case. The latter limit is incorporated into the mean-field
equations by taking the boundary condition ␩s = 0, used previously in Refs. [7,10].
Our assumptions for treating the adsorbed layer are as
follows. (i) Inside the adsorption layer, the electrostatic interactions are assumed to be stronger than the excludedvolume interactions. (ii) Using results from Ref. [10], we
assume that the electrostatic potential decays mainly via the
PE adsorption and not via the salt (for weak enough ionic
strength). (iii) Another assumption is that the electrostatic
potential ␨ ⬅ e␺ / kBT can be written in terms of a scaling
function ␨ = 兩␨s兩h共x / D兲, where 兩␨s兩 is the surface potential and
D is the adsorption layer length scale (see also Ref. [10]).
(iv) The last assumption is that the electrostatic potential ␨ is
low enough so that we can employ the linear Debye-Hückel
approximation.
terms on the right-hand side (RHS) are the ideal gas pressure
of the salt and counterions. The third term is the pressure due
to the interaction between the electrostatic field and the
monomer concentration, and the last term is the excludedvolume driven pressure.
For every segment where ␨ is a monotonic function of x,
a change of variables from x to ␨ can be performed. Using
d␩ / dx = 共d␩ / d␨兲共d␨ / dx兲 in Eq. (11) yields
冉
1−
2 2
a
km
关␩⬘共␨兲兴2
3f
冊冉 冊
d␨
dx
2
2 ␨
= 2␬2共cosh ␨ − 1兲 + 2km
共e − 1
1
2
共␩2 − 1兲2 .
− ␨␩2兲 − v␾2bkm
f
共12兲
Below, for the strong repulsive case, we attempt to produce
approximate solutions to Eqs. (11) and (12), and compare
them to numerical calculations of Eqs. (3) and (4) (see also
Ref. [10]).
Under the above assumptions, the excluded-volume term
in Eq. (12) can be neglected, and exp共␨兲 and cosh共␨兲 can be
expanded out to second order in ␨, yielding
冉 冊
d␨
dx
2
−
冉 冊冉 冊
2 2
km
a d␩
3f d␨
d␨
dx
2
2
2
= ␬2␨2 + 2km
␨共1 − ␩2兲 + km2 ␨2 .
共13兲
The first term on the RHS relates to the salt ions, the second
to both the monomer and counterion concentrations, and the
third term to the counterion concentration.
A. First integration of the mean-field equations
B. Scaling of potential and concentration profiles
The salt dependence of the adsorption characteristic can
be obtained from the first integration of Eqs. (3) and (4).
Multiplying Eq. (3) by d␨ / dx and Eq. (4) by d␩ / dx, and then
integrating both equations from x to infinity yields
Under the above assumptions, the dominant term in the
RHS of Eq. (13) is the second one related to the monomer
and counterion contributions. This term changes sign from
negative to positive at ␩ ⯝ 1. For ␩ → 0 (close to the surface)
the negative sign of the RHS implies that the second (nega2 2
a . Fitive) term of the LHS is dominant: d␩ / d␨ ⬎ 冑3f / km
nally, it can be shown self-consistently (presented below)
that the first term on the LHS is of order 兩␨s兩3 while the
correction terms on the RHS are of order 兩␨s兩2. Hence close to
the surface we neglect the first term in the LHS,
冉 冊
1 d␨
2 dx
2
冉
2
= ␬2共cosh ␨ − 1兲 + km
e␨ − 1 +
冕
⬁
x
␩2
冊
d␨
dx ,
dx
共9兲
冉 冊
a2 d␩
12 dx
2
1
= v␾2b共␩2 − 1兲2 − f
4
冕
⬁
x
␩␨
d␩
dx.
dx
2
Equation (10) is then multiplied by 2km
/ f and subtracted
from Eq. (9).
Using 兰⬁x 共d␨ / dx兲␩2dx = −␩2␨ − 2兰⬁x 共d␩ / dx兲␨␩dx we get
冉 冊
1 d␨
2 dx
2
−
冉 冊
2 2
km
a d␩
6f dx
2
2 ␨
2
= ␬2共cosh ␨ − 1兲 + km
共e − 1兲 − km
␨␩2
−
1
v␾2k2 共␩2 − 1兲2 ,
2f b m
冉 冊冉 冊
共10兲
2 2
km
a d␩
3f d␨
d␨
dx
2
2
2
= 2km
兩␨兩共1 − ␩2兲 − 共km
+ ␬2兲␨2 . 共14兲
On the charged surface ␩ = 0 and ␨ = −兩␨s兩. Using the
mean-value theorem in the interval of interest 0 艋 ␩ 艋 1 we
get 兩d␩ / d␨兩y=ys ⯝ 1 / 兩␨s兩. Substituting the scaling hypothesis
d␨ / dx 兩 x=0 ⯝ 兩␨s兩 / D in Eq. (14) evaluated at the surface yields
an estimate for the layer thickness D:
共11兲
which can be interpreted as the local pressure balance equation. The first term on the left-hand side (LHS) of Eq. (11) is
the electrostatic field pressure, and the second term is the
pressure arising from chain elasticity. The first and second
2
a
D⯝
冑6f兩␨s兩
⯝
冑6f兩␨s兩
a
冉
冉
1−
␬2 + km2
2 兩 ␨ s兩
2km
冊
−1/2
冊
1
csalt
1 + 兩 ␨ s兩 +
兩 ␨ s兩 .
4
2f ␾2b
共15兲
The first term of Eq. (15) retrieves the result given in Refs.
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A. SHAFIR AND D. ANDELMAN
[7,10,12,14]. The other terms include corrections due to the
ionic strength of the solution. As salt is added, the monomermonomer electrostatic repulsion wins over the electrostatic
attraction of the PE’s to the surface, resulting in an increase
in the adsorption length D, and at very high salt in depletion
[10].
By changing x to the dimensionless length x / D, neglecting the excluded-volume term and inserting the potential
scaling hypothesis in Eq. (4), the scaling form for the monomer concentration ␾2b␩2 can be derived. The scaling form of
␨ [see assumption (iii) above] dictates a similar scaling form:
␩ = ␾共x兲 / ␾b ⯝ 冑␾2M / ␾2b g共x / D兲, where ␾2M is the peak monomer concentration, and g is a scaling function normalized to
one at the peak and satisfies g共0兲 = 0.
To find ␾2M , the peak concentration condition d␩ / d␨ = 0
can be inserted in Eq. (13), causing the second term on the
LHS to vanish. Using the scaling for ␨ and D of Eq. (15)
yields
冉
␾2M = ␾2b +
3兩␨s兩2
4 ␲ l Ba 2
冊冉
1−
2csalt + f ␾2b
2f ␾2b
冊
兩 ␨ s兩 ,
共16兲
where factors depending on the value of h, the scaling function for the potential, and its derivative evaluated at the peak
position are omitted for clarity. The above equation is in
agreement with previous results [7,10] calculated in the limit
of no added salt and negligible effect of counterions. As the
amount of salt increases, the monomer concentration characterized by ␾2M decreases, and for large enough amount of
added salt, the peak monomer concentration decreases below
its bulk value—a clear sign of depletion.
The amount of adsorbed monomers in the adsorption
layer is now calculated as a function of ionic strength,
冋
⌫ ⯝ ␾2M,0D0 1 −
冉
2csalt + f ␾2b
2
1
兩 ␨ s兩 2 + 2
4f
␾b ␾M,0
冊册
, 共17兲
where the added subscript zero denotes the known no-salt
limits, ␾2M,0 = 3兩␨s兩2 / 共4␲lBa2兲 and D0 = a / 冑6f兩␨s兩, of the maximal monomer concentration ␾2M and adsorption length D,
respectively [7,10]. For no added salt, the adsorbed amount
scales like
⌫ = ⌫0 ⯝ ␾2M,0D0 ⬃ 兩␨s兩3/2 f
−1/2 −1 −1
lB a .
共18兲
When salt is added, the surface potential screening is obtained via the PE’s and salt ions. In addition, the adsorbed
amount ⌫ decreases as can be seen from the negative correction term in Eq. (17).
Using the above assumptions, the scaling of the induced
surface charge is
兩␴兩 = 共4␲lB兲−1
冏 冏
d␨
dx
⬃
x=0
兩␨s兩3/2 f 1/2
兩 ␨ s兩
⬃
.
4 ␲ l BD
l Ba
共19兲
Comparing this scaling to Eq. (17), we see that the scaling of
␴ resembles that of the charge carried by the adsorbed
amount f⌫. Subtracting the two equations shows that the
overcharging ⌬␴ = f⌫ − ␴ scales like ␴ as well.
The numerically calculated salt dependence of the adsorption length D is presented in Fig. 4(a). The location of the
concentration peak, taken as D, is shown as a function of the
bulk salt concentration csalt. The length D is shown to increase with the addition of salt, in agreement with Eq. (15)
and with experimental results [26,30]. The salt dependence
of the overall adsorbed amount ⌫ ⬅ ␾2b兰⬁0 共␩2 − 1兲dx is presented in Fig. 4(b). The adsorbed amount is shown to decrease steadily with the addition of salt, in agreement with
Eqs. (17). The sharp drop in ⌫ is a sign of PE depletion in
high salt conditions, and shows that the higher terms in the
⌫共csalt兲 expansion are negative as well.
We close this section by mentioning that a more elaborated treatment of the overcharging for strongly repulsive
surfaces is presented in the Appendix. We find two different
scaling regimes: one for electrostatically dominated overcharging and the second for the excluded volume dominated
one. For the electrostatically dominated regime the overcharging follows the same scaling as the overall PE adsorption, ⌬␴ ⬃ ⌫ ⬃ f ␴. In the excluded volume dominated regime
its scaling depends on the excluded volume parameter and
yields ⌬␴0 ⯝ f兩␴兩冑lBa2 / 共v2␾2b兲. In both cases the magnitude
of overcharging is very small as compared to the bare ␴. This
weak overcharging is not sufficient to explain PE multilayer
formation for the repulsive surfaces considered in this section. Note that a very different situation exists when the surfaces are chemically attractive as discussed in Sec. III.
V. WEAK CHEMICALLY REPULSIVE SURFACES
In Sec. IV, we treated the strong chemically repelling surface, and concluded that the PE-surface electrostatic attraction is the sole drive for the adsorption. In this section, we
relax the assumption of strong repulsive surfaces, while preserving the dominance of the electrostatic interactions.
The chemical interactions are added into our model via
the amount of PE in direct contact with the adsorbing surface
␾共x = 0兲 ⬅ ␾s [or in the renormalized form ␩共x = 0兲 = ␩s]. We
note that the case of weakly repulsive surfaces the slope of
the monomer order parameter near the surface must be positive, 兩d␩ / dx兩x=0 ⬎ 0, as explained in Sec. II.
We begin by noting that the mean-field equations (3) and
(4) are invariant to translations in the coordinate x. Namely,
the equations are invariant under a translational transformation x → x + l, as long as the boundary conditions are also
transformed in the same manner, i.e., ␾共l兲 = ␾s and
␨共l兲 = −兩␨s兩. We note that this is a property of the planar geometry used in this case, and that for different geometries
such as spherical or cylindrical this shift symmetry is no
longer present.
Using the above symmetry, the adsorption profile characterized by ␾共x兲, ␨共x兲, with boundary conditions ␾共0兲 = ␾s,
␨共0兲 = −兩␨s兩, can be thought of as part of a larger profile, satisfying ␨共−l兲 ⬅ −兩␨s*兩 and ␾共−l兲 = 0 at the x = −l phantom surface. Such a profile, in turn, is exactly similar to the one
discussed in the previous subsection, since the two systems
are connected by the above mentioned translational transformation. Therefore finding the above ␨s* and l from the given
061804-6
POLYELECTROLYTE ADSORPTION: CHEMICAL AND …
PHYSICAL REVIEW E 70, 061804 (2004)
FIG. 4. (a) The width of the concentration profile D, taken as the peak location, is presented as a function of csalt, the added-salt
concentration. The dotted line corresponds to f = 0.1, the dashed line to f = 0.18, the dashed-dotted line to f = 0.56, and the solid line to f
= 1. Other parameters used are ␧ = 80, T = 300 K, 兩␨s兩 = 1, a = 5 Å, ␾2b = 10−6 Å−3, v = 50 Å3. The length scale of the adsorption D is seen to
increase with the addition of salt. For high enough added salt concentrations, the concentration peak vanishes altogether, indicating that the
polymer is depleted from the surface. The adsorption-depletion crossover is denoted by a full circle. (b) The total adsorbed amount ⌫
= 兰⬁0 dx共␾2 − ␾2b兲 ⬃ cmD is plotted against the amount of added salt. The solid line corresponds to f = 0.03, the dashed line to f = 0.1, the
dash-dot to f = 0.31 and the dotted line to f = 1. The adsorbed amount decreases slowly with salt for low amounts of added salt. For high
concentrations of added salt the adsorbed amount decreases sharply to negative values, signaling an adsorption-depletion transition. Other
parameters used are the same as in (a) (reproduced from Ref. [10]).
␾s, 兩␨s兩 then enables the derivation of the adsorption characteristics in the same manner as discussed in Sec. IV.
This strategy is demonstrated in Fig. 5. In Fig. 5(a) the
monomer concentration profiles are presented as a function
of the distance from the surface x for several surface parameters 0 ⬍ ␾s ⬍ 16.2␾b and 0 ⬍ 兩␨s兩 ⬍ 1.0. The profiles were obtained by solving Eqs. (3) and (4) numerically using the
boundary conditions ␩共0兲 = ␾s / ␾b, ␨共0兲 = −兩␨s兩 close to the
surface and ␨共x → ⬁兲 = 0, ␩共x → ⬁兲 = 1 at infinity. In Fig. 5(b)
the same profiles are shown, with a translation in the surface
position. As can be seen in Fig. 5(b), all three profiles collapse on exactly the same profile, showing that all three profiles have the same concentration and length scales despite
the difference in 兩␨s兩 and ␩s. These scales can be calculated
by using Eqs. (15) and (16), as discussed in the previous
subsection.
Using the scaling relations Eqs. (15) and (16), the above
symmetry yields the following connections between 兩␨s*兩, l
and the surface boundary conditions 兩␨s兩 and ␾s:
冉 冊
共20兲
冉 冊
共21兲
␾s = ␾*M g
␨s = 兩␨s*兩h
l
,
D*
l
,
D*
where D*, ␾*M are the same as Eqs. (15) and (16) when we
change 兩␨s兩 → 兩␨s*兩. Manipulation of Eq. (21) then yields
兩␨s*兩 =
␨s
␨s
.
* =
−1
h共l/D 兲 h„g 共␾s/␾*M 兲…
共22兲
Equation (22) can be solved iteratively for 兩␨s*兩 potential as a
function of the surface and bulk solution parameters. Insertion of the result into Eq. (21) then gives l / D*.
It is important to note that the inversion of Eq. (21) is
only possible in the region where ␾ and x are monotonic,
i.e., when the surface monomer concentration is lower than
the maximal monomer concentration on the profile ␾s2
2
⬍ ␾max
. This condition can also be expressed by the condition 兩d␩ / dx兩x=0 ⬎ 0, showing that the chemical interactions
between the surface and the PE chains must still be repulsive
for the shift strategy to be employed.
We now proceed to solve Eq. (22) for low values of
␾s / ␾M . Using the boundary conditions and Eqs. (3) and (4)
the characteristic functions h共x⬘兲, g共x⬘兲 can be expanded in
powers of x⬘ to yield g共x⬘兲 = a0x⬘ + b0x⬘3 + ¯ and h共x⬘兲 = −1
+ a1x⬘ + b1x⬘2 + ¯. Inserting these expansions into Eq. (22)
and solving to second order in ␾s yields
冋
兩␨s*兩 = 兩␨s兩 1 +
册
a1 ␾s
+ ¯.
a0 ␾ M
共23兲
Note that ␾2M is now no longer the actual maximal monomer
concentration, but is defined in Eq. (16) as the characteristic
for the monomer concentration, where the real surface electrostatic potential 兩␨s兩 is used rather than the phantom surface
potential.
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A. SHAFIR AND D. ANDELMAN
FIG. 5. (a) The numerically calculated monomer concentration ␩2共x兲 is plotted as a function of the distance from the charged surface x,
for several values of 兩␨s兩 and ␩s ⬅ ␾s / ␾b. The solid line corresponds to ␩s = 0 and 兩␨s兩 = 1.0, the dashed line with triangle markers to ␩s
= 12.55 and 兩␨s兩 = 0.6, and the dashed-dotted line with square markers to ␩s = 16.2 and 兩␨s兩 = 0.34. All profiles share csalt = 0.01M, f = 1, a
= 5 Å, v = 10 Å3, ␾2b = 10−6 Å−3, ␧ = 80, and T = 300 K. All profiles are found to have the same height in the peak monomer concentration,
despite the difference in the boundary conditions. (b) The same profiles as in part (a), after a shift in the x = 0 position is used. The solid line
is not shifted, the dashed line with triangular markers is shifted by ⌬x = 2.35 Å, and the dashed-dotted line with square markers is shifted by
⌬x = 4.12 Å. All three profiles, show a data collapse on the solid line profile, corresponding to 兩␨s*兩 = 1.0.
We can now turn to calculate the overall adsorption. Returning to the initial assumption that the monomer concentration profile is a part of a larger profile starting at x = −l, the
adsorbed amount can be taken as the total adsorbed amount
from x = −l to infinity, minus the amount adsorbed from
x = −l to the actual surface at x = 0:
⌫=
冕
⬁
dx共␾2 − ␾2b兲 =
0
冕
⬁
dx共␾2 − ␾2b兲 −
−l
冕
0
dx共␾2 − ␾2b兲.
−l
共24兲
The first integral in the RHS of Eq. (24) is the overall adsorption of PE to the phantom surface at x = −l, while the
second integral is the PE adsorbed amount between this
phantom surface and the real surface at x = 0. Using the expansion of g共x兲 and Eq. (20), we can see that the second
integral is of third order in ␾s / ␾ M , and can be neglected for
low enough ␾s / ␾ M . Using Eqs. (17) and (20), Eq. (24) can
be evaluated as
冑3兩␨s兩3/2
⌫⯝
4␲冑2lBaf 1/2
冉
⫻ 1+
冋
1−
2csalt + f ␾2b
2csalt + f ␾2b
12f ␾2b
4f ␾2b
兩 ␨ s兩
冊册
.
strongly repulsive surface case. In both cases, when the
amount of salt increases to a high enough value, higher order
terms become dominant, and the adsorbed amount decreases,
similar to the infinitely repulsive case in Fig. 4(b).
The salt dependence of the adsorbed amount ⌫ is shown
for several ␾s values in Fig. 6(a). The adsorbed amount is
shown to always increase with the amount of monomers attached to the surface ␾s2. For low ␾s values, the adsorbed
amount decreases with salt, as expected from Eq. (25) and in
agreement with Fig. 4(b). For higher ␾s values, the adsorbed
amount is seen to increase for low bulk concentrations of salt
and then decrease strongly. In Fig. 6(b) the relative overcharging is plotted for the same ␾s values, showing that the
relative overcharging is still a very small effect, even for
weakly repulsive surfaces. This shows that the attractive
chemical interactions between the surface and the PE chains
are indeed crucial for the multilayer formation, and not the
electrostatic interactions alone.
VI. DISCUSSION AND CONCLUSIONS
3a1␾s
兩 ␨ s兩 +
2a0␾ M,0
共25兲
As expected, the adsorbed amount increases with ␾s. However, the salt dependence of the surface is very different from
the strongly repulsive surface case. For low amounts of
added salt and high enough ␾s, the term combining the small
ions and ␾s is stronger than the salt term, and the adsorbed
amount increases with salt. When ␾s is low, the addition of
salt causes the adsorbed amount to decrease, similar to the
We have presented analytical and numerical calculations
of the mean-field equations describing adsorption of polyelectrolytes onto charged surfaces. Three surface situations
are discussed: chemically attractive surfaces, and strong and
weak chemical repulsive ones.
The strongest adsorption phenomenon is seen from numerical solution of the mean-field equations for chemically
attractive surfaces. This manifests itself in a large overcharging of about 40–55 % of the bare surface charge for high salt
conditions. On the other hand, we did not find a full charge
inversion as was predicted earlier in Ref. [6]. This means that
any model [27,28] which tries to explain multilayer forma-
061804-8
POLYELECTROLYTE ADSORPTION: CHEMICAL AND …
PHYSICAL REVIEW E 70, 061804 (2004)
FIG. 6. (a) Numerically calculated adsorbed amount of monomers ⌫ is plotted against the added salt concentration csalt for several values
of ␩s = ␾s / ␾b, for the case of weak chemically repulsive surfaces. The solid line corresponds to ␩s = 5, the dashed line corresponds to ␩s
= 50, and the dashed-dotted line to ␩s = 100. All profiles share 兩␨s兩 = 1.0, f = 1.0, a = 5 Å, v = 10 Å3, ␾2b = 10−8 Å−3, ␧ = 80, and T = 300 K. The
adsorbed amount is seen to increase with ␩s for all values of csalt. For low amounts of added salt, the adsorbed amount is seen to increase
with salt, in contrast to the ␩s = 0 case (strong repulsive surface) in Fig. 4(b), while for high amounts of added salt the adsorbed amount
decreases, in agreement with Fig. 4(b). (b) The relative overcharging ⌬␴ / 兩␴兩 is plotted against the amount of added salt for the same
parameters as in part (a). Despite the increase in the adsorbed amount, the relative overcharging remains a very small effect. At high salt
concentration, the relative overcharging becomes negative—signaling an undercompensation of the surface charge.
tion at equilibrium needs to rely on nonelectrostatic complexation between the cationic and anionic polymer chains,
beside the electrostatic interactions. So far no simple scaling
forms are obtained for the PE adsorbed amount in this case.
However, the adsorbed amount (and the overcharging) are
shown to increase with the salt amount csalt, and to decrease
with f, the charge fraction on the PE chain as well as with d,
which is inversely proportional to the chemical interaction of
the surface.
For the case of strong chemical repulsion between the
surface and the PE chains, we find a difference in the effect
of salt addition on the width of the adsorbed layer D and on
the adsorbed PE amount ⌫ (Fig. 4). The width of the PE
adsorbed layer increases with the addition of salt, while the
overall adsorbed PE amount decreases with the addition of
salt. This difference results from a strong decrease in the
monomer concentration close to the surface upon the addition of salt. This difference between ⌫ and D is in agreement
with experimental results of Shubin et al. [30], where silica
surfaces were used to adsorb cationic polyacrylamide
(CPAM). When salt is further added to the solution, the PE’s
stop overcompensating the charged surface, and we are in an
undercompensation regime of adsorption. Previous studies
[10] showed that for even higher amounts of salt, scaling as
csalt ⬃ f兩␾s兩, the PE’s deplete from the charged surface. For
such surfaces, the overcharging in most cases is found to
scale like the induced surface charge density ␴. For weakly
charged PE’s the overcharging depends on the excludedvolume parameter and bulk monomer concentration, and
scales as ⬃f ␴. Very weakly charged PE’s are shown to adsorb to the charged surface, but do not overcharge it. Our
scaling results are in agreement with numerical calculations
of the mean-field equations. For all PE charges, the overcharging of the PE with respect to a repelling surface is
found to be very small, of the order of 1% of the bare surface
charge. This naturally leads to the conclusion that the overcharging relies heavily on the chemical interactions between
the surface and the PE chains. It is of much smaller importance for this type of repulsive surfaces than for attractive
ones considered above.
For weakly repulsive surfaces, we find that the adsorbed
amount increases with the addition of salt for low salt
amounts, and decreases for high amounts of added salt. This
is a natural interpolation between the attractive and repulsive
surface limits. Moreover, the overcharging of the surface
charge remains low, and for high amounts of added salt the
PE again undercompensates the surface charges.
Our results can serve as a starting point for a more quantitative analysis of the overcharging phenomena, and provide
for better understanding of multilayer formation.
ACKNOWLEDGMENTS
We thank I. Borukhov, Y. Burak, M. Castelnovo, J.-F.
Joanny, and E. Katzav for useful discussions and comments.
In particular, we are indebted to Qiang (David) Wang for his
critical comments on a previous version of this manuscript.
Support from the Israel Science Foundation (ISF) under
Grant No. 210/01 and the U.S.-Israel Binational Foundation
(BSF) under Grant No. 287/02 is gratefully acknowledged.
061804-9
PHYSICAL REVIEW E 70, 061804 (2004)
A. SHAFIR AND D. ANDELMAN
FIG. 7. (a) Numerically calculated excess adsorption ⌬⌫, defined as the PE adsorbed amount from the potential peak to infinity, is plotted
against f. The squares correspond to 兩␨s兩 = 0.5, and the triangles to 兩␨s兩 = 0.6. Both profiles share ␾2b = 10−6 Å−3, v = 102Å3, a = 5 Å, csalt
= 0.1 mM, T = 300 K, and ␧ = 80. The two profiles can be fitted in the low f region by ⌬⌫ ⬃ f 1/2 (dashed line), followed by a high f region
where ⌬⌫ ⬃ f −1/2 (solid line). These scaling results are in agreement with Eqs. (A3) and (A6). (b) ⌬⌫ is plotted against the surface potential
兩␨s兩. The squares correspond to f = 0.1, and the triangles to f = 0.3. All other parameters are the same as in (a). The two profiles show a scaling
of ⌬⌫ ⬃ 兩␨s兩3/2, fitted by a solid line, in agreement with Eqs. (A3) and (A6). The constant prefactors in the fitting lines are obtained by
imposing the condition that the fitting line passes through the last numerical data point in the respective regime.
冏 冉 冊冏
a2 d␩
6 dx
APPENDIX: SCALING OF THE OVERCHARGING LAYER
IN CHEMICALLY REPULSIVE SURFACES
In order to derive scaling estimates for the PE adsorption
regime, the adsorbed PE layer is divided into two sublayers
around the potential peak point x ⬅ xc. The compensation
layer is defined as x ⬍ xc, and consists of PE’s attracted to the
surface mainly by electrostatics. In contrast, in the overcharging layer 共x ⬎ xc兲 the PE chains are electrostatically repelled from the surface, but remain in the surface vicinity
solely because of their chain connectivity. For low enough
salt concentrations, the amount of charge carried by the PE’s
in the adsorbed layer is much larger than that carried by the
small ions. In this case, the overcharging can be taken as
⌬␴ ⬅ − 兩␴兩 + f ␾2b
冕
⬁
0
dx共␩2 − 1兲 ⯝ f ␾2b
冕
⬁
dx共␩2 − 1兲,
xc
共A1兲
where the first integral is the same as Eq. (7), with ␴ being
the induced surface charge on the adsorbing surface. The
second integral in the above equation is taken from x = xc to
infinity, and describes the PE adsorption in the region where
the electrostatic interaction between the PE and the surface is
repulsive. The integral from x = 0 to xc balances the surface
charge almost entirely for low amounts of added salt, using
the fact that 兩d␨ / dx兩xc = 0 (Gauss law).
In order to find the amount of polymers in the overcharging layer, we begin by examining Eq. (11) at the potential
peak, x = xc, where the first term on the LHS vanishes. Expanding the RHS to second order in ␨ without neglecting the
excluded-volume term yields
2
1
= f ␨c共␩2c − 1兲 + v␾2b共␩2c − 1兲2
2
x=xc
−
冉 冊
f ␬2
2
2 + 1 ␨c ,
2 km
共A2兲
where the values at the peak are denoted by ␨c ⬅ ␨共xc兲 and
␩c ⬅ ␩共xc兲. As shown below, ␩c decreases with addition of
salt. Therefore, for a large amount of added salt, the LHS of
Eq. (A2) becomes negative (this is true to all orders of ␨c and
not only to order ␨2c as shown here), while the RHS is always
positive, meaning that there is no peak in the rescaled potential ␨. This demonstrates that surface charge overcharging
can only occur in low enough salt conditions.
In the overcharging layer, the previous assumptions made
in Sec. III A about the dominance of the electrostatic interactions are not necessarily true. Consequently, the decay of
the PE concentration in this region can be governed by either
one of two interactions: the electrostatic or the excludedvolume repulsion between the monomers. We consider them
as two limits for overcharging. (i) An electrostatically dominated regime, v␾2b共␩2c − 1兲 Ⰶ f ␨c, where the excluded-volume
term is neglected in Eq. (A2). (ii) The excluded-volume
dominated regime, v␾2b共␩2c − 1兲 Ⰷ f ␨c. Here, the electrostatic
interaction between monomers in Eq. (A2) is neglected. In
both regimes the value of d␩ / dx at the peak position is estimated from the scaling of the compensation layer to be
兩d␩ / dx兩x=xc ⯝ ␾ M / 共␾bD兲.
1. Strongly charged regime: v␾2b„␩2c − 1… ™ f␨c
The validity criterion of this regime is similar to the assumptions presented in Sec. IV A, showing that the over-
061804-10
POLYELECTROLYTE ADSORPTION: CHEMICAL AND …
PHYSICAL REVIEW E 70, 061804 (2004)
charging layer 共x ⬎ xc兲 can be thought of as an extension of
the compensation layer 共0 ⬍ x ⬍ xc兲. Instead of rederiving ⌬␴
from Eq. (A2), we can use Eq. (14) and the expressions for D
and ␾2M from Eqs. (15) and (16), which yield ␨c ⬃ 兩␨s兩 and
␩2c ⬃ ␾2M / ␾2b. The resulting overcharging scales like the bare
surface charge:
⌬␴ ⬃ f⌫D ⬃ f共␾2M − ␾2b兲D ⬃ 兩␴兩.
共A3兲
Note that the effect of added salt in this regime is similar to
the one described in Eq. (17). Namely, the added salt lowers
the surface charge overcharging, in agreement with experimental [30] and numerical [10] results.
2. Intermediate charged regime: v␾2b„␩2c − 1… š f␨c
In this regime, the excluded-volume interactions dominate
the decay of the PE concentration. Using the regime validity
condition and neglecting the first term on the RHS of Eq.
(A2) yields an expression for the monomer concentration at
x c:
␩2c = 1 +
冑 冏冉 冊 冏
d␩
dx
a2
3v␾2b
2
+
x=xc
␬2 + km2 2
f ␨c .
2
km
v␾2b
共A4兲
Noting that the d␩ / dx is continuous at x = xc, we use the
compensation layer scaling estimates to find 兩d␩ / dx兩x=xc
⯝ ␾ M / 共␾bD兲. Substituting the latter into Eq. (A4) and expanding to first order in both the ionic strength 2csalt + f ␾2b
2
= 共␬2 + km
兲 / 4␲lB and the ratio of the bulk and peak monomer
concentrations ␾2b / ␾2M ⯝ lBa2␾2b / 兩␨s兩2 yields
␩2c ⯝ 1 +
冑
冋
册
2csalt + f ␾2b
2␲lBa2␾2b
3 兩␨s兩3/2 f 1/2
1
−
兩
␨
兩
+
,
s
2
2␲ 冑lBa2v␾2b
3兩␨s兩2
2f ␾b
共A5兲
␾2M ,
respecwhere Eqs. (15) and (16) are used for D and
tively.
The overcharging is calculated from Eq. (7) where the
characteristic length scale entering the integral is the Edwards length, ␰e = a / 冑3v␾2b, depending only on the excludedvolume interactions and not on the salt. The overcharging
⌬␴ ⯝ f ␾2b共␩2c − 1兲␰e is
⌬␴ ⯝
兩 ␨ s兩
3/2 3/2
f
冑2␲lB␾2bv
冋
1−
2csalt + f ␾2b
兩 ␨ s兩
2f ␾2b
2␲lBa ␾2b
3兩␨s兩2
2
+
册
.
共A6兲
In the limit of no ionic strength (csalt = 0 and negligible counterion contribution), the overcharging from Eq. (A6) scales
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[2]
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like ⌬␴0 ⯝ f兩␴兩冑lBa2 / 共v2␾2b兲. Addition of salt results in a decrease of overcharging, similar to what was shown in the
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We define the adsorption excess ⌬⌫ by integrating numerically the adsorbed amount from xc to ⬁. Note that in the
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7(b), ⌬⌫ is presented as a function of 兩␨s兩. The scaling ⌬⌫
⬃ 兩␨s兩3/2, also shown in the figure, is in agreement with Eqs.
(18), (A3), and (A6). We note, by comparing Fig. 4(b) to Fig.
7, that the overcompensating PE’s are of the order of less
than 1% of the total adsorbed amount, showing that the
mean-field overcharging of repulsive surfaces is an extremely small effect.
3. Undercompensation threshold
So far we presented the case where the PE layer is overcompensating the surface charge, and discussed it in two
limits of strong and intermediate charged PE’s. In some
range of system parameters the PE charges do not overcompensate the surface ones. The threshold for having this undercompensation is briefly discussed here.
Re-examining the validity of Eq. (A2), it can be seen that
for high enough salt the third term on the right dominates the
largest of the first two terms when
f兩␨s兩
1
csalt + f ␾2b ⬎
2
l Ba 2
共A7兲
up to some numerical pre-factors. Note that the same inequality is valid in both limits of strong and intermediate
regimes, as long as we are in high salt conditions. It should
be noted that a similar scaling rule was found in Refs.
[1,2,10,14] for the adsorption-depletion crossover. Numerical
results show that the overcharging-undercompensation transitions indeed have the same scaling, but differ in the constant multiplying the scaling result.
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